prompt stringlengths 0 256 | answer stringlengths 1 256 |
|---|---|
matrix elements computed with vanishing lepton mass can be absorbed into this distribution function, which can be calculated in perturbation theory. Representative diagrams for the photon-initiated processes are shown in Fig.~\ref{fig:NLOphotdiag}.
\beg | in{figure}[h]
\centering
\includegraphics[width=3.0in]{NLOphot.pdf}%
\caption{Representative Feynman diagrams contributing to the $q(p_1)+\gamma(p_2) \to q(p_3)+g(p_4)$ (left) and $g(p_1)+\gamma(p_2) \to q(p_3)+\bar{q}(p_4)$ scattering processes.} \label{f |
ig:NLOphotdiag}
\end{figure}
The full expression for the NLO hadronic cross section then takes the form
\begin{eqnarray}\label{eq:hxsecnlo}
{\rm d}\sigma_{\rm NLO}&=&\int\frac{{\rm d}\xi_1}{\xi_1}\frac{{\rm d}\xi_2}{\xi_2}\bigg\{f_{g/H}^1f_{l/l}^2 {\rm | d}\hat{\sigma}_{gl}^{(2,1)} + f_{g/H}^1f_{\gamma/l}^2{\rm d}\hat{\sigma}_{g\gamma}^{(1,1)} \phantom{\sum_q}\nonumber \\
&& +\sum_q\bigg[f_{q/H}^1f_{l/l}^2{\rm d}\hat{\sigma}_{ql}^{(2,1)}+f_{\bar{q}/H}^1f_{l/l}^2{\rm d}\hat{\sigma}_{\bar{q}l}^{(2,1)} \nonu |
mber \\
&&+f_{q/H}^1f_{\gamma/l}^2{\rm d}\hat{\sigma}_{q\gamma}^{(1,1)}+f_{\bar{q}/H}^1f_{\gamma/l}^2{\rm d}\hat{\sigma}_{\bar{q}\gamma}^{(2,1)}\bigg]\bigg\},
\end{eqnarray}
where we have abbreviated $f_{i/j}^k=f_{i/j}(\xi_k)$. The contributions ${\rm d}\ | hat{\sigma}_{gl}^{(2,1)}$, ${\rm d}\hat{\sigma}_{ql}^{(2,1)}$ and ${\rm d}\hat{\sigma}_{\bar{q}l}^{(2,1)}$ denote the usual DIS partonic channels computed to NLO in QCD with zero lepton mass. The terms ${\rm d}\hat{\sigma}_{\bar{q}\gamma,\rm LO}$ and ${\r |
m d}\hat{\sigma}_{g\gamma,\rm LO}$ denote the new contributions arising when $Q^2 \approx 0$ and the virtual photon is nearly on-shell. The photon distribution can be expressed as
\begin{equation}
f_{\gamma/l}(\xi) = \frac{\alpha}{2\pi} P_{\gamma l}(\xi) | \left[ \text{ln}\left(\frac{\mu^2}{\xi^2 m_l^2} \right)-1\right] +{\cal O}(\alpha^2),
\end{equation}
where the splitting function is given by
\begin{equation}
\label{eq:gammalsplit}
P_{\gamma l}(\xi) = \frac{1+(1-\xi)^2}{\xi}.
\end{equation}
This is the w |
ell-known Weizs\"{a}cker-Williams (WW) distribution for the photon inside of a lepton~\cite{WW}. The appearance of the renormalization scale $\mu$ indicates that an $\overline{\text{MS}}$ subtraction of the QED collinear divergence is used in the calculat | ion of the $gl$ and $ql$ scattering channels, and consequently in the derivation of the photon distribution function.
\section{Calculation of the NNLO result} \label{sec:nnlo}
The calculation of the full ${\cal O}(\alpha^2\alpha_s^2)$ corrections involv |
es several distinct contributions. The quark-lepton and gluon-lepton scattering channels receive two-loop double virtual corrections, one-loop corrections to single real-emission diagrams, and double-real emission corrections. These contributions necessi | tate the use of a full-fledged NNLO subtraction scheme. We use the recently-developed $N$-jettiness subtraction scheme~\cite{Boughezal:2015dva,Gaunt:2015pea}. Its application to this process is discussed here in detail. In addition, the photon-initiated |
scattering channels receive virtual and single real-emission corrections. The calculation of these terms follows the standard application of the antennae subtraction scheme at NLO.
There is in addition a new effect that appears at the NNLO level. The | initial-state lepton can emit a photon which splits into a $q\bar{q}$ pair, all of which are collinear to the initial lepton direction. In the limit of vanishing fermion masses there is a collinear singularity associated with this contribution. This dive |
rgence appears in the quark-lepton, gluon-lepton, and photon-initiated scattering channels. It can be absorbed into a distribution function that describes the quark distribution inside a lepton. Treating the collinear singularity in this way leads to new | scattering channels that first appear at NNLO: $q\bar{q} \to q\bar{q}$, $q\bar{q} \to q^{\prime}\bar{q}^{\prime}$, $q\bar{q} \to gg$, $qq^{\prime} \to qq^{\prime}$, and $qg \to qg$. For our numerical predictions for these channels we need the quark distr |
ibution in a lepton. We obtain this by solving the DGLAP equation, which to the order we are working can be written as
\begin{eqnarray}\label{eq.dglap}
&&\mu^2\frac{\partial f_{q/l}}{\partial \mu^2}(\xi,\mu^2)=e_q^2\frac{\alpha}{2\pi}\int_{\xi}^1\frac{{\r | m d} z}{z}P_{q\gamma}^{(0)}(z)\,f_{\gamma/l}\left(\frac{\xi}{z},\mu^2\right) \nonumber \\ &&+ e_q^2\left(\frac{\alpha}{2\pi}\right)^2\int_{\xi}^1\frac{{\rm d} z}{z}P_{ql}^{(1)}(z)\,f_{l/l}\left(\frac{\xi}{z},\mu^2\right),
\end{eqnarray}
where the two neede |
d splitting kernels are
\begin{eqnarray}
P_{q\gamma}^{(0)}(x)&=&x^2+(1-x)^2, \nonumber \\
P_{ql}^{(1)}(x) &=& -2+\frac{20}{9x}+6x-\frac{56x^2}{9}\nonumber \\ &&\hspace{-0.8cm}+\left(1+5x+\frac{8x^2}{3}\right) \log(x)-(1+x) \log^2(x).
\end{eqnarray}
This ex | pression for the NLO splitting kernel can be obtained from upon replacement of the appropriate QCD couplings with electromagnetic ones. To derive the full quark-in-lepton distribution we use as an initial condition $ f_{q/l}(\xi,m_l^2)=0$. Solving Eq.~(\r |
ef{eq.dglap}) with this initial condition gives
\begin{eqnarray}
\label{eq:qinl}
&&f_{q/l}(\xi,\mu^2)=e_q^2\left(\frac{\alpha}{2\pi}\right)^2\bigg\{\bigg[\frac{1}{2}+\frac{2}{3\xi}-\frac{\xi}{2}-\frac{2\xi^2}{3} \nonumber \\ && +(1+\xi)\log\xi \bigg] \log | ^2\left(\frac{\mu^2}{m_l^2}\right) + \bigg[-3-\frac{2}{\xi}+7\xi-2\xi^2+\nonumber \\ &&\bigg( -5 -\frac{8}{3\xi}+\xi+\frac{8\xi^2}{3}\bigg)\log(\xi)-3(1+x)\log^2(\xi)\bigg] \nonumber \\ && \times \log\left(\frac{\mu^2}{m_l^2}\right).
\end{eqnarray}
\begi |
n{figure}[h]
\centering
\includegraphics[width=3.0in]{NNLO2jet.pdf}%
\caption{Representative Feynman diagrams contributing to the $q(p_1)+\bar{q}(p_2) \to g(p_3)+g(p_4)$ (left), $q(p_1)+q^{\prime}(p_2) \to q(p_3)+q^{\prime}(p_4)$ (middle), and $q(p_1)+g(p_ | 2) \to q(p_3)+g(p_4)$ (right) scattering processes.} \label{fig:NNLO2jet}
\end{figure}
With this distribution function it is straightforward to obtain numerical predictions for these partonic channels. Representative Feynman diagrams for these processes |
are shown in Fig.~\ref{fig:NNLO2jet}. We can now write down the full result for the ${\cal O}(\alpha^2 \alpha_s^2)$ correction to the cross section:
\begin{eqnarray}
\label{eq:hxsecnnlo}
{\rm d}\sigma_{\rm NNLO}&=&\int\frac{{\rm d}\xi_1 {\rm d}\xi_2}{\xi_ | 1 \xi_2}\bigg\{f_{g/H}^1f_{l/l}^2\,{\rm d}\hat{\sigma}_{gl}^{(2,2)}
+f_{g/H}^1f_{\gamma/l}^2\,{\rm d}\hat{\sigma}_{g\gamma}^{(1,2)} \phantom{\sum_q}\nonumber \\
&+&\sum_q\bigg[f_{g/H}^1f_{q/l}^2\,{\rm d}\hat{\sigma}_{gq}^{(0,2)}+f_{g/H}^1f_{\bar{q}/l}^2\,{ |
\rm d}\hat{\sigma}_{g\bar{q}}^{(0,2)}\bigg] \nonumber\\
&+&\sum_q\bigg[f_{q/H}^1f_{l/l}^2\,{\rm d}\hat{\sigma}_{ql}^{(2,2)}+f_{\bar{q}/H}^1f_{l/l}^2\,{\rm d}\hat{\sigma}_{\bar{q}l}^{(2,2)} \nonumber \\
&+&f_{q/H}^1f_{\gamma/l}^2\,{\rm d}\hat{\sigma}_{q\gam | ma}^{(1,2)}+f_{\bar{q}/H}^1f_{\gamma/l}^2\,{\rm d}\hat{\sigma}_{\bar{q}\gamma}^{(1,2)} \phantom{\sum_q\bigg[} \nonumber \\
&+&f_{q/H}^1f_{\bar{q}/l}^2\,{\rm d}\hat{\sigma}_{q\bar{q}}^{(0,2)}+f_{\bar{q}/H}^1f_{q/l}^2\,{\rm d}\hat{\sigma}_{\bar{q}q}^{(0,2)} |
\bigg] \phantom{\sum_q} \nonumber \\
&+&\sum_{q,q'}\bigg[f_{q/H}^1f_{q'/l}^2\,{\rm d}\hat{\sigma}_{qq'}^{(0,2)}+f_{\bar{q}/H}^1f_{\bar{q}'/l}^2\,{\rm d}\hat{\sigma}_{\bar{q}\bar{q}'}^{(0,2)} \nonumber \\
&+&f_{q/H}^1f_{\bar{q}'/l}^2\,{\rm d}\hat{\sigma}_{ | q\bar{q}'}^{(0,2)}+f_{\bar{q}/H}^1f_{q'/l}^2\,{\rm d}\hat{\sigma}_{\bar{q}q'}^{(0,2)}\bigg]\bigg\}. \phantom{\sum_q}
\end{eqnarray}
The most difficult contribution is the quark-lepton scattering channel. It receives contributions from two-loop virtual c |
orrections (double-virtual), one-loop corrections to single real emission terms (real-virtual), and double-real emission corrections. Sample Feynman diagrams for these corrections are shown in Fig.~\ref{fig:NNLOql}. These are separately infrared divergen | t, and require a full NNLO subtraction scheme to combine. We apply the $N$-jettiness subtraction scheme~\cite{Boughezal:2015dva,Gaunt:2015pea}. The starting point of this method is the $N$-jettiness event shape variable~\cite{Stewart:2010tn}, defined in |
the one-jet case of current interest as
\begin{equation}
\label{eq:taudef}
{\cal T}_1 = \frac{2}{Q^2} \sum_i \text{min}\left\{p_B \cdot q_i, p_J \cdot q_i \right\},
\end{equation}
with $Q^2 = -(p_2-p_4)^2$. Here, $p_B$ and $p_J$ are light-like four-vector | s along the initial-state hadronic beam and final-state jet directions, respectively.\footnote{This choice of ${\cal T}_1$ corresponds to $\tau_1^a$ in Ref.~\cite{Kang:2013nha}. We note that this definition is dimensionless, unlike the choice in previous |
applications of $N$-jettiness subtraction~\cite{Boughezal:2015dva}.} The $q_i$ denote the four-momenta of all final-state partons. Values of ${\cal T}_1$ near zero indicate a final state containing a single narrow energy deposition, while larger values | denote a final state containing two or more well-separated energy depositions. A measurement of ${\cal T}_1$ is itself of phenomenological interest. It has been proposed as a probe of nuclear properties in electron-ion collisions~\cite{Kang:2012zr,Kang:2 |
013wca}, and has also been suggested as a way to precisely determine the strong coupling constant~\cite{Kang:2013nha}.
\begin{figure}[h]
\centering
\includegraphics[width=3.0in]{NNLOql.pdf}%
\caption{Representative Feynman diagrams contributing to the qua | rk-lepton scattering channel at NNLO.} \label{fig:NNLOql}
\end{figure}
We will use ${\cal T}_1$ to establish the complete ${\cal O}(\alpha^2\alpha_s^2)$ calculation of the quark-lepton scattering channel. Our ability to do so relies on two key observa |
tions, as first discussed in Ref.~\cite{Boughezal:2015dva} for a general $N$-jet process.
\begin{itemize}
\item Restricting ${\cal T}_1 >0$ removes all doubly-unresolved limits of the quark-lepton matrix elements, for example when the two additional part | ons that appear in the double-real emission corrections are soft or collinear to the beam or the final-state jet. This can be seen from Eq.~(\ref{eq:taudef}); if ${\cal T}_1 >0$ then at least one $q_i$ must be resolved. Since all doubly-unresolved limits |
are removed, the ${\cal O}(\alpha^2\alpha_s^2)$ correction in this phase space region can be obtained from an NLO calculation of two-jet production in electron-nucleon collisions.
\item When ${\cal T}_1$ is smaller than any other hard scale in the prob | lem, it can be resummed to all orders in perturbation theory~\cite{Stewart:2009yx,Stewart:2010pd}. Expansion of this resummation formula to ${\cal O}(\alpha^2\alpha_s^2)$ gives the NNLO correction to the quark-lepton scattering channel for small ${\cal T} |
_1$.
\end{itemize}
The path to a full NNLO calculation is now clear. We partition the phase space for the real-virtual and double-real corrections
into regions above and below a cutoff on ${\cal T}_1$, which we label ${\cal T}_1^{cut}$:
\begin{equatio | n}
\label{eq:partition}
\begin{split}
{\rm d}\sigma_{ql}^{(2,2)} &= \int {\rm d}\Phi_{\text{VV}} \, |{\cal M}_{\text{VV}}|^2 +\int {\rm d}\Phi_{\text{RV}} \, |{\cal M}_{\text{RV}}|^2 \, \theta_1^{<} \\
&+\int {\rm d}\Phi_{\text{RR}} \, |{\cal M}_{\text{RR} |
}|^2 \, \theta_1^{<}+\int {\rm d}\Phi_{\text{RV}} \, |{\cal M}_{\text{RV}}|^2 \, \theta_1^{>} \\
&+\int {\rm d}\Phi_{\text{RR}} \, |{\cal M}_{\text{RR}}|^2 \, \theta_1^{>} \\
\equiv & {\rm d}\sigma_{ql}^{(2,2)}({\cal T}_1 < {\cal T}_1^{cut})+{\rm d}\sigma | _{ql}^{(2,2)}({\cal T}_1 > {\cal T}_1^{cut})
\end{split}
\end{equation}
We have abbreviated $\theta_1^{<} = \theta({\cal T}_1^{cut}-{\cal T}_1)$ and $\theta_1^{>} = \theta({\cal T}_1-{\cal T}_1^{cut})$. The first three terms in this expression all have ${ |
\cal T}_1<{\cal T}_1^{cut}$, and have been collectively denoted as ${\rm d}\sigma_{ql}^{(2,2)}({\cal T}_1 < {\cal T}_1^{cut})$. The remaining two terms have ${\cal T}_1>{\cal T}_1^{cut}$, and have been collectively denoted as ${\rm d}\sigma_{ql}^{(2,2)}({ | \cal T}_1 > {\cal T}_1^{cut})$. The double-virtual corrections necessarily have ${\cal T}_1=0$. We obtain ${\rm d}\sigma_{ql}^{(2,2)}({\cal T}_1 > {\cal T}_1^{cut})$ from a NLO calculation of two-jet production. This is possible since no genuine double- |
unresolved limit occurs in this phase-space region. We discuss the calculation of ${\rm d}\sigma_{ql}^{(2,2)}({\cal T}_1 < {\cal T}_1^{cut})$ using the all-orders resummation of this process in the following sub-section. We note that only the quark-lept | on and gluon-lepton scattering channels have support for ${\cal T}_1=0$. For the other processes there are two final-state jets with non-zero transverse momentum. Such configurations necessarily have ${\cal T}_1 >0$, and therefore only receive contributi |
ons from the
above-the-cut phase-space region. We only need these contributions to at most NLO in QCD perturbation theory, which can be obtained via standard techniques. We use antennae subtraction. The only non-standard aspect of this NLO calculation | is the appearance of triple-collinear QED limits associated with the emission of a photon and a $q\bar{q}$ pair which require the use of integrated antennae found in Refs.~\cite{Daleo:2009yj,Boughezal:2010mc,GehrmannDeRidder:2012ja}. A powerful aspect of |
the $N$-jettiness subtraction method is its ability to upgrade existing NLO calculations to NNLO precision. Previous applications of $N$-jettiness subtraction~\cite{Boughezal:2015dva,Boughezal:2015aha,Boughezal:2015ded} have used the NLO dipole subtracti | on technique~\cite{Catani:1996vz} to facilitate the calculation of the above-cut phase-space region. This work demonstrates that it can also be used in conjunction with the NLO antennae subtraction scheme.
\subsubsection{Below ${\cal T}_1^{cut}$}
An |
all-orders resummation of the ${\cal T}_1$ event-shape variable in the DIS process for the limit ${\cal T}_1 \ll 1$ was constructed in Refs.~\cite{Kang:2013wca,Kang:2013nha}:
\begin{eqnarray}
\label{eq:SCETfac}
\frac{{\rm d} \sigma}{{\rm d} {\cal T}_1} &= | & \int {\rm d} \Phi_2(p_3,p_4;p_1,p_2) \int {\rm d} t_J {\rm d} t_B {\rm d} k_S \nonumber \\ &\times & \delta\left( {\cal T}_1-\frac{t_J}{Q^2}-\frac{t_B}{Q^2} -\frac{k_S}{Q} \right) \nonumber \\ &\times & \sum_q J_q(t_J,\mu) \,S(k_S,\mu)
H_q(\Phi_2,\mu |
) B_q(t_B,x,\mu)+ \ldots \nonumber \\
\end{eqnarray}
We have allowed the index $q$ to run over both quarks and anti-quarks. $x$ denotes the usual Bjorken scaling variable for DIS:
\begin{equation}
x = \frac{Q^2}{2 P\cdot (p_2-p_4)},
\end{equation}
where | $P$ is the initial-state nucleon four-momentum. $\Phi_2$ denotes the Born phase space, which consists of a quark and a lepton. The derivation of this result relies heavily on the machinery of soft-collinear effective theory (SCET)~\cite{Bauer:2000ew}. |
A summary of the SCET functions that appear in this expression and what they describe is given below.
\begin{itemize}
\item $H$ is the hard function that encodes the effect of hard virtual corrections. At leading order in its $\alpha_s$ expansion it redu | ces to the leading-order partonic cross section. At higher orders it also contains the finite contributions of the pure virtual corrections, renormalized using the $\overline{\text{MS}}$ scheme. It depends only on the Born-level kinematics and on the sca |
le choice.
\item $J_q$, the quark jet function, describes the effect of radiation collinear to the final-state jet (which for this process is initiated by a quark at LO). It depends on $t_J$, the contribution of final-state collinear radiation to ${\cal | T}_1$. It possesses a perturbative expansion in $\alpha_s$.
\item $S$ is the soft function that encodes the contributions of soft radiation. It depends on $k_S$, the contribution of soft radiation to ${\cal T}_1$, and has a perturbative expansion in $\ |
alpha_s$.
\item $B$ is the beam function that contains the effects of initial-state collinear radiation. It depends on $t_B$, the contribution of initial-state collinear radiation to ${\cal T}_1$. The beam function is non-perturbative; however, up to co | rrections suppressed by $\Lambda^2_{\text{QCD}}/t_B$, it can be written as a convolution of perturbative matching coefficients and the usual PDFs:
\begin{equation}
B_q(t_B,x,\mu) = \sum_i \int_x^1 \frac{{\rm d} \xi}{\xi} {\cal I}_{qi} (t_B,x/\xi,\mu) f_{i/ |
H}(\xi),
\end{equation}
where we have suppressed the scale dependence of the PDF, and $i$ runs over all partons.
\end{itemize}
The delta function appearing in Eq.~(\ref{eq:SCETfac}) combines the contribution of each type of radiation to produce the measur | ed value of ${\cal T}_1$. The ellipsis denotes power corrections that are small as long as we restrict ourselves to the phase-space region ${\cal T}_1 \ll 1$.
The hard, jet, and soft functions as well as the beam-function matching coefficients all have |
perturbative expansions in $\alpha_s$ that can be obtained from the literature~\cite{Zijlstra:1992qd,Becher:2006qw,Gaunt:2014xga,Boughezal:2015eha}. Upon expansion to ${\cal O}(\alpha_s^2)$ and integration over the region ${\cal T}_1 < {\cal T}_1^{cut}$, | Eq.~(\ref{eq:SCETfac}) will give exactly the cross section ${\rm d}\sigma_{ql}^{(2,2)}({\cal T}_1 < {\cal T}_1^{cut})$ that we require. We can match the beam function onto the PDFs and rewrite the cross section below the cut as
\begin{eqnarray}
{\rm d}\s |
igma &=& \int \frac{{\rm d} \xi_1}{\xi_1} \frac{{\rm d} \xi_2}{\xi_2} \sum_{q,i} f_{i/H}(\xi_1) f_{l/l}(\xi_2) \int {\rm d} \Phi_2(p_3,p_4;p_1,p_2) \nonumber \\ && \int_0^{{\cal T}_1^{cut}} {\rm d} {\cal T}_1 \int {\rm d} t_J {\rm d} t_B {\rm d} k_S \de | lta\left( {\cal T}_1-\frac{t_J}{Q^2}-\frac{t_B}{Q^2} -\frac{k_S}{Q} \right) \nonumber \\ &\times & J_q(t_J,\mu) \,S(k_S,\mu)
H_q(\Phi_2,\mu) \, {\cal I}_{qi} (t_B,x/\xi,\mu) \nonumber \\
&\equiv& \sum_{q,i} \int {\rm d}\Phi^{i}_{\text{Born}} \, \left[ J |
_q \otimes S \otimes H_q \otimes {\cal I}_{qi} \right].
\end{eqnarray}
We have introduced the schematic notation $J_q \otimes S \otimes H_q \otimes {\cal I}_{qi}$ for the integrations of the SCET functions over ${\cal T}_1$, $t_J$, $t_B$, and $k_S$; ${\rm | d}\Phi^{i}_{\text{Born}}$ represents all other terms for the given index $i$: the parton distribution functions, the integral over the Born phase space, and any measurement function acting on the Born variables. We denote the expansion of these functions |
in $\alpha_s$ as
\begin{equation}
{\cal X} = {\cal X}^{(0)}+{\cal X}^{(1)}+{\cal X}^{(2)}+\ldots,
\end{equation}
where the superscript denotes the power of $\alpha_s$ appearing in each term. With this notation, we need the following contributions to obtai | n the ${\cal O}(\alpha_s^2)$ correction to the cross section below ${\cal T}_1^{cut}$:
\begin{eqnarray}
\label{eq:belowcutexp}
{\rm d}\sigma_{\text{NNLO}} &=& \sum_{q,i} \int {\rm d}\Phi^i_{\text{Born}} \, \bigg\{J_q^{(2)} \otimes S^{(0)} \otimes H_q^{(0) |
} \otimes {\cal I}_{qi}^{(0)} \nonumber \\
&+&J_q^{(0)} \otimes S^{(2)} \otimes H_q^{(0)} \otimes {\cal I}_{qi}^{(0)} + J_q^{(0)} \otimes S^{(0)} \otimes H_q^{(2)} \nonumber \\ &\otimes &
{\cal I}_{qi}^{(0)}+J_q^{(0)} \otimes S^{(0)} \otimes H_q^{(0)} \ | otimes {\cal I}_{qi}^{(2)} \nonumber \\
&+&J_q^{(1)} \otimes S^{(1)} \otimes H_q^{(0)} \otimes {\cal I}_{qi}^{(0)}+J_q^{(1)} \otimes S^{(0)} \otimes H_q^{(1)} \nonumber \\ &\otimes & {\cal I}_{qi}^{(0)}
+J_q^{(1)} \otimes S^{(0)} \otimes H_q^{(0)} \otim |
es {\cal I}_{qi}^{(1)}+J_q^{(0)} \otimes S^{(1)} \nonumber \\ & \otimes &H_q^{(1)} \otimes {\cal I}_{qi}^{(0)}
+ J_q^{(0)} \otimes S^{(1)} \otimes H_q^{(0)} \otimes {\cal I}_{qi}^{(1)}\nonumber \\ &+& J_q^{(0)} \otimes S^{(0)} \otimes H_q^{(1)} \otime | s {\cal I}_{qi}^{(1)} \bigg\}.
\end{eqnarray}
To simplify this expression, we first note that the hard function has no dependence on the hadronic variables ${\cal T}_1$, $t_J$, $t_B$, and $k_S$. It depends only on the Born phase space and is a multiplica |
tive factor for the hadronic integrations. Next, we note that the leading-order expressions for the SCET functions are proportional to delta functions in their respective hadronic variable:
\begin{equation}
{\cal X}^{(0)} \propto \delta(t_{\cal X})
\end{e | quation}
for ${\cal X}=J_q, S$, or ${\cal I}_{qi}$. This simplifies the integrals involving an ${\cal X}^{(2)}$. Using the $J_q^{(2)}$ term in Eq.~(\ref{eq:belowcutexp}) as an example, we have
\begin{equation}
J_q^{(2)} \otimes S^{(0)} \otimes H_q^{(0)} |
\otimes {\cal I}_{qi}^{(0)} = Q^2 H_q^{(0)} \delta_{qi}\, \int_0^{{\cal T}_1^{cut}} \hspace{-0.5cm}{\rm d} {\cal T}_1 J_q^{(2)} ({\cal T}_1 Q^2,\mu),
\end{equation}
where the $\delta_{qi}$ comes from the ${\cal I}_{qi}^{(0)}$ term. Using the fact that the | jet function can be written in the form
\begin{equation}
J_q^{(2)} (t_J,\mu) = a_{-1} \delta(t_J)+\sum_{n=0}^3 a_n \, \frac{1}{\mu^2} \left[ \frac{\mu^2 \,\text{ln}^n(t_J/\mu^2)}{t_J}\right]_+,
\end{equation}
where the $a_i$ denote coefficients that can |
be found in the literature~\cite{Becher:2006qw}, we can immediately derive
\begin{equation}
\begin{split}
J_q^{(2)} \otimes S^{(0)} \otimes H_q^{(0)} & \otimes {\cal I}_{qi}^{(0)} = H_q^{(0)} \,\delta_{qi} \left\{ a_{-1} \right. \\ & \left. +\sum_{n=0}^3 | \frac{1}{n+1} a_{n+1} \,\text{ln}^{n+1} \left( \frac{{\cal T}_1^{cut} Q^2}{\mu^2}\right)\right\}.
\end{split}
\end{equation}
We have used the standard definition of the plus distribution of a function:
\begin{equation}
\int_0^1 {\rm d} x \, \left[f(x)\ri |
ght]_+ \, g(x) = \int_0^1 {\rm d} x \, f(x) \left[g(x)-g(0) \right].
\end{equation}
This analysis shows how to analytically calculate any term containing one of the NNLO SCET functions.
It remains only to calculate contributions containing two NLO SCET | functions. We focus on $J_q^{(1)} \otimes S^{(1)} \otimes H_q^{(0)} \otimes {\cal I}_{qi}^{(0)} $ as an example. Using the LO expression for the beam function we can immediately derive the equation
\begin{equation}
\begin{split}
J_q^{(1)} \otimes S^{(1)} |
\otimes H_q^{(0)} &\otimes {\cal I}_{qi}^{(0)} = Q \, H_q^{(0)} \,\delta_{qi}\, \int_0^{{\cal T}_1^{cut}} \hspace{-0.5cm}{\rm d} {\cal T}_1 \int_0^{{\cal T}_1 Q} dt_J \\ & \times
J_q^{(1)} (t_J,\mu) \, S^{(1)}\left(Q{\cal T}_1-\frac{t_J}{Q} \right).
\end | {split}
\end{equation}
The NLO results for the jet and soft functions can be written as
\begin{eqnarray}
J_q^{(1)} (t_J,\mu) &=& b_{-1} \delta(t_J)+\sum_{n=0}^1 b_n \, \frac{1}{\mu^2} \left[ \frac{\mu^2 \,\text{ln}^n(t_J/\mu^2)}{t_J}\right]_+, \nonumber \ |
\
S^{(1)} (k_S,\mu) &=& c_{-1} \delta(k_S)+\sum_{n=0}^1 c_n \, \frac{1}{\mu} \left[ \frac{\mu \,\text{ln}^n(k_S/\mu)}{k_S}\right]_+ .
\end{eqnarray}
Using these expressions it is straightforward to derive the result
\begin{eqnarray}
J_q^{(1)} &\otimes& S | ^{(1)} \otimes H_q^{(0)} \otimes {\cal I}_{qi}^{(0)} = H_q^{(0)} \,\delta_{qi} \bigg\{ b_{-1} c_{-1}
\nonumber \\ &+& c_{-1} \, \sum_{n=0}^1 \frac{1}{n+1} \,b_{n+1}\, L_J^{n+1}
+ b_{-1} \, \sum_{n=0}^1 \frac{1}{n+1} \,c_{n+1}\, L_S^{n+1}
\nonumber \\ &+ |
&\left. \sum_{n,m=0}^1 b_m c_n \left[ L_J^{\alpha} \,L_S^{\beta} \,\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(1+\alpha+\beta)}\right] \bigg|_{\alpha^m,\beta^n}\right\}.
\end{eqnarray}
The vertical bar in the last term indicates that we should take the $\alp | ha^m \beta^n$ coefficient of the series expansion of the bracketed term. We have introduced the abbreviations
\begin{equation}
L_J = \text{ln} \left( \frac{{\cal T}_1^{cut} Q^2}{\mu^2}\right), \;\;\; L_S = \text{ln} \left( \frac{{\cal T}_1^{cut} Q}{\mu}\r |
ight).
\end{equation}
Using these results it is straightforward to analytically compute all of the necessary hadronic integrals in Eq.~(\ref{eq:belowcutexp}). The remaining integrals are over the Born phase space and parton distribution functions, and are | simple to perform numerically. This completes the calculation of the ${\cal T}_1 < {\cal T}_1^{cut}$ phase space region. We note that the cross section below ${\cal T}_1^{cut}$ will contain terms of the form $\text{ln}^n ({\cal T}_1^{cut})$, where $n$ |
ranges from 0 to 4 at NNLO. An important check of this framework is the cancellation of these terms against the identical logarithms that appear for ${\cal T}_1 > {\cal T}_1^{cut}$. We must also choose ${\cal T}_1^{cut}$ small enough to avoid the power c | orrections in Eq.~(\ref{eq:SCETfac}) that go as ${\cal T}_1^{cut}/Q$. Both of these issues will be addressed in our later section on numerical results.
\subsubsection{Above ${\cal T}_1^{cut}$}
We now briefly outline the computation of the quark-lepton ch |
annel in the phase space region ${\cal T}_1 > {\cal T}_1^{cut}$. As discussed above this can be obtained from a NLO calculation of the DIS process with an additional jet. In addition to the virtual corrections to the $q\,l \to q\,l\,g$ partonic process | , there are numerous radiation processes that also contribute: $q\,l \to q\,l\,g\,g$, $q\,l \to q\,l\,q'\,\bar{q}'$ and $q\,l \to q\,l\,q\,\bar{q}$. In addition to the usual ultraviolet renormalization of the strong coupling constant, the real and virtua |
l corrections are separately infrared divergent. Remaining divergences after introducing an NLO subtraction scheme are associated with initial-state collinear singularities, and are handled via
mass factorization.
\section{Validation and numerical resul | ts} \label{sec:numerics}
We have implemented the NNLO cross section of Eq.~(\ref{eq:hxsecnnlo}), as well as the LO and NLO results of Eqs.~(\ref{eq:sigLO}) and~(\ref{eq:hxsecnlo}) in a numerical code {\tt DISTRESS} that allows for arbitrary cuts to be imp |
osed on the final-state lepton and jets. We describe below the checks we have performed on our calculation.
The antennae subtraction method provides an analytic cancellation of the $1/\epsilon$ poles that appear in an NLO calculation. We are therefore a | ble to check this cancellation of poles for all components computed in this way. This includes the entire $\sigma_{\rm NLO}$, as well as the following contributions to the NNLO hadronic cross section: ${\rm d}\hat{\sigma}_{g\gamma}^{(1,2)}$ and ${\rm d}\h |
at{\sigma}_{q\gamma}^{(1,2)}$. The various contributions ${\rm d}\hat{\sigma}_{ij}^{(0,2)}$ that occur in $\sigma_{\rm NNLO}$ are finite, and simple to obtain. Our NLO results for the transverse momentum distribution and pseudorapidity distribution of th | e leading jet are compared against the plots of Ref.~\cite{Hinderer:2015hra}. We find good agreement with these results.
This leaves only the validation of the ${\rm d}\hat{\sigma}_{gl}^{(2,2)}$ and ${\rm d}\hat{\sigma}_{ql}^{(2,2)}$ contributions, which |
both utilize the $N$-jettiness subtraction technique. There are two primary checks that these pieces must satisfy. First, they must be independent of the parameter ${\cal T}_N^{cut}$. This checks the implementation of the beam, jet and soft functions, | which have logarithmic dependence on this parameter. It also determines the range of ${\cal T}_N^{cut}$ for which the power corrections denoted by the ellipsis in Eq.~(\ref{eq:SCETfac}) are negligibly small. Second, upon integration over the final-state |
hadronic phase space we must reproduce the NNLO structure functions first determined in Ref.~\cite{Zijlstra:1992qd}. This is an extremely powerful check on our calculation, which essentially cannot be passed if any contribution is implemented incorrectly. |
We show in Fig.~\ref{fig:valid} the results of these checks for the $ql$ and $gl$ scattering channels. We have set the total center-of-mass energy of the lepton-proton collision to 100 GeV. For the purpose of this validation check only we have imposed |
the phase-space cut $Q^2>100$ GeV$^2$, and have integrated inclusively over the hadronic phase space. We have equated the renormalization and factorization scales to the common value $\mu=Q$, and have used the CT14 NNLO parton distribution functions~\cite | {Dulat:2015mca}. ${\cal T}_1^{cut}$ has been varied from $5 \times 10^{-6}$ to $1 \times 10^{-4}$, and the ratio of the NNLO correction to the LO result for the cross section is shown. The solid lines show the prediction of the inclusive structure functi |
on. We first note that correction is extremely small, less than 1\% of the leading-order result. Nevertheless we have excellent numerical control over the NNLO coefficient, as indicated by the vertical error bars. Our numerical error on the NNLO coeffic | ient is at the percent-level, sufficient for 0.01\% precision on the total cross section. The $N$-jettiness prediction for the $ql$ scattering channel is independent of ${\cal T}_1^{cut}$ over the studied range, while the $gl$ scattering channel is indepe |
ndent of ${\cal T}_1^{cut}$ for ${\cal T}_1^{cut} < 10^{-5}$. Both channels are in excellent agreement with the structure-function result. We have also checked bin-by-bin that the transverse momentum and pseudorapidity distributions of the jet have no dep | endence on ${\cal T}_1^{cut}$.
\begin{figure}[h]
\centering
\includegraphics[width=3.6in]{valid.pdf}%
\caption{Plot of the NNLO corrections normalized to the LO cross section for the quark-lepton and gluon-lepton scattering channels as a function of ${\ |
cal T}_1^{cut}$. The points denote values obtained from $N$-jettiness subtraction, with the vertical error bars denoting the numerical errors, while the solid lines indicate the inclusive structure function result. } \label{fig:valid}
\end{figure}
Havin | g established the validity of our calculation we present phenomenological results for proposed EIC run parameters. We set the collider energy to $\sqrt{s}=100$ GeV and study the inclusive-jet transverse momentum and pseudorapidity distributions in the ran |
ge $p_T^{jet}>5$ GeV and $|\eta_{jet}|<2$. We use the CT14 parton distribution set~\cite{Dulat:2015mca} extracted to NNLO in QCD perturbation theory. We reconstruct jets using the anti-$k_t$ algorithm~\cite{Cacciari:2008gp} with radius parameter $R=0.5$. | Our central scale choice for both the renormalization and factorization scales is $\mu=p_T^{jet}$. To estimate the theoretical errors from missing higher-order corrections we vary the scale around its central value by a factor of two. The transverse mo |
mentum distributions at LO, NLO and NNLO are shown in Fig.~\ref{fig:ptj}. The $K$-factors, defined as the ratios of higher order over lower order cross sections, are shown in the lower panel of this figure. The NNLO corrections are small in the region $p | _T^{jet}>10$ GeV , changing the NLO result by no more than 10\% over the studied $p_T^{jet}$ region. The shift of the NLO cross section is slightly positive in the low transverse momentum region, and become less than unity at high-$p_T^{jet}$. Both the N |
NLO corrections and the scale dependence grow large at low-$p_T^{jet}$. The large scale dependence arises primarily from the partonic channels $qq$ and $gq$. These channels are effectively treated at leading order in our calculation, since they first app | ear at ${\cal O}(\alpha^2\alpha_s^2)$, and they are evaluated at the low scale $\mu=p_T^{jet}/2$ in our estimate of the theoretical uncertainty. It is therefore not surprising that their uncertainty dominates at low-$p_T^{jet}$ These channels do not cont |
ribute at NLO, and consequently the NLO scale uncertainty is smaller. This is an example of the potential pitfalls in using the scale uncertainty as an estimate of the theoretical uncertainty. Only an explicit calculation can reveal qualitatively new eff | ects that occur at higher orders in perturbation theory.
\begin{figure}[h]
\centering
\includegraphics[width=3.65in]{ptj.pdf}%
\caption{Plot of the inclusive-jet transverse momentum distribution at LO, NLO and NNLO in QCD perturbation theory. The upper |
panel shows the distributions with scale uncertainties, while the lower panel shows the $K$-factors for the central scale choice.} \label{fig:ptj}
\end{figure}
We next show the pseudorapidity distribution in Fig.~\ref{fig:etaj}, with the restriction $p_T | ^{jet}>10$ GeV. There are a few surprising aspects present in the NNLO corrections. First, the scale dependence at NNLO in the region $\eta_{jet}<0$ is larger than the corresponding NLO scale variation. Although the corrections are near unity over most |
of the studied pseudorapidity range, they become sizable near $\eta_{jet} \approx 2$, reducing the NLO rate by nearly 50\%. To determine the origin of these effects we show in Fig.~\ref{fig:eta_breakdown_muv} the breakdown of the NNLO correction into its | separate partonic channels. This reveals that the total NNLO correction comes from an intricate interplay between all contributing channels, with different ones dominating in different $\eta_{jet}$ regions. Only the gluon-lepton partonic process is neglig |
ible over all of phase space. For negative $\eta_{jet}$, the dominant contribution is given by the quark-quark process. As discussed before, this appears first at ${\cal O}(\alpha^2\alpha_s^2)$. It is therefore effectively treated at leading-order in ou | r calculation, and consequently has a large scale dependence. We note that the quark-in-lepton distribution from Eq.~(\ref{eq:qinl}) is larger at high-$x$ than the corresponding photon-in-lepton one, leading to this channel being larger in the negative $\ |
eta_{jet}$ region. At high $\eta_{jet}$, the distribution receives sizable contributions from the gluon-photon process. No single partonic channel furnishes a good approximation to the shape of the full NNLO correction.
\begin{figure}[h]
\centering
\inc | ludegraphics[width=3.65in]{etaj.pdf}%
\caption{Plot of the inclusive-jet pseudorapidity distribution at LO, NLO and NNLO in QCD perturbation theory. The upper panel shows the distributions with scale uncertainties, while the lower panel shows the $K$-fact |
ors for the central scale choice.} \label{fig:etaj}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=3.65in]{eta_breakdown_muv.pdf}%
\caption{Breakdown of the NNLO correction to the $\eta_{jet}$ distribution into its constituent partonic | channels, as a ratio to the full NLO cross section in the bin under consideration. Also shown is the total result obtained by summing all channels. The bands indicate the scale variation. } \label{fig:eta_breakdown_muv}
\end{figure}
\section{Conclusions |
} \label{sec:conc}
We have presented in this paper the full calculation of the ${\cal O}(\alpha^2 \alpha_s^2)$ perturbative corrections to jet production in electron-nucleus collisions. To obtain this result we have utilized the $N$-jettiness subtraction | scheme introduced to allow NNLO calculations in hadronic collisions. We have described the application of this method to inclusive jet production in detail, and have shown that upon integration over the final-state hadronic phase that we reproduce the k |
nown NNLO result for the inclusive structure functions. Our results have been implemented in a numerical program {\tt DISTRESS} that we plan to make publicly available for future phenomenological studies.
We have shown numerical results for jet productio | n at a proposed future EIC. Several new partonic channels appear at the ${\cal O}(\alpha^2 \alpha_s^2)$ level, which have an important effect on the kinematic distributions of the jet. No single partonic channel furnishes a good approximation to the full |
NNLO result. The magnitude of the corrections we find indicate that higher-order predictions will be an important part of achieving the precision understanding of proton structure desired at the EIC, and we expect that the methods described here will be a | n integral part of achieving this goal.
\vspace{0.5cm}
\noindent
\mysection{Acknowledgements}
G.~A. is supported by the NSF grant PHY-1520916. R.~B. is supported by the DOE contract DE-AC02-06CH11357. X.~L. is supported by the DOE grant DE-FG02-93ER-407 |
62. F.~P. is supported by the DOE grants DE-FG02-91ER40684 and DE-AC02-06CH11357. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U. | S. Department of Energy under Contract No. DE-AC02-05CH11231. It also used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. This research was supported in p |
art by the NSF under Grant No. NSF PHY11-25915 to the Kavli Institute of Theoretical Physics in | Santa Barbara, which we thank for hospitality during the completion of this manuscript.
|
\section{Introduction}
\label{sec:intro}
Compact stars have a large number of pulsation modes that have been extensively studied since the seminal work of Chandrasekhar on radial oscillations \cite{APJ140:417:1964,PRL12:114:1964}. In general, these modes | are very difficult to observe in the electromagnetic spectrum; therefore most efforts have concentrated on gravitational wave asteroseismology in order to characterise the frequency and damping times of the modes that emit gravitational radiation. In part |
icular, various works focused on the oscillatory properties of pure hadronic stars, hybrid stars and strange quark stars trying to find signatures of the equation of state of high density neutron star matter (see \cite{AA325:217:1997,IJMPD07:29:1998,AA36 | 6:565:2001,APJ579:374:2002,PRD82:063006:2010,EL105:39001:2014} and references therein).
More recently, compact star oscillations have attracted the attention in the context of Soft Gamma ray Repeaters (SGRs), which are persistent X-ray emitters that spora |
dically emit short bursts of soft $\gamma$-rays. In the quiescent state, SGRs have an X-ray luminosity of $\sim 10^{35}$ erg/s, while during the short $\gamma$-bursts they release up to $10^{42}$ erg/s in episodes of about 0.1 s. Exceptionally, some of | them have emitted very energetic giant flares which commenced with brief $\gamma$-ray spikes of $\sim 0.2$ s, followed by tails lasting hundreds of seconds. Hard spectra (up to 1 MeV) were observed during the spike and the hard X-ray emission of the tail g |
radually faded modulated at the neutron star (NS) rotation period. The analysis of X-ray data of the tails of the giant flares of SGR 0526-66, SGR 1900+14 and SGR 1806-20 revealed the presence of quasi-periodic oscillations (QPOs) with frequencies ranging | from $\sim$ 18 to 1840 Hz \cite{APJ628:L53:2005,APJ632:L111:2005,AA528:A45:2011}. There are also candidate QPOs at higher frequencies up to $\sim 4$ kHz in other bursts but with lower statistical significance \cite{ElMezeini2010}; in fact, according to a |
more recent analysis only one burst shows a marginally significant signal at a frequency of around 3706 Hz \cite{Huppenkothen2013}.
Several characteristics of SGRs are usually explained in terms of the \textit{magnetar} model, assuming that the object | is a neutron star with an unusually strong magnetic field ($B \sim 10^{15} $ G) \cite{Woods2006}. In particular, giant flares are associated to catastrophic rearrangements of the magnetic field. Such violent phenomena are expected to excite a variety of o |
scillation modes in the stellar crust and core. In fact, recent studies have accounted for magnetic coupling between the crust and the core, and associate QPOs to global magneto-elastic oscillations of highly magnetized neutron stars \cite{Levin2007,Cerda | Duran2009,Colaiuda2012,Gabler2014}. There has also been interest in the possible excitation of low order $f$-modes because of their strong coupling to potentially detectable gravitational radiation \cite{Levin2011}.
In the present paper we focus on radi |
al oscillations of neutron stars permeated by ultra-strong magnetic fields. These modes might be relevant within the magnetar model because they could be excited during the violent events associated with gamma flares. Since they have higher frequencies tha | n the already known QPOs, they cannot be directly linked to them at present. However, it is relevant to know all the variety of pulsation modes of strongly magnetized neutron stars because the number of observations is still small and new features could em |
erge in future flares' data.
On the other hand, in the case of rotating objects we can expect some amount of gravitational radiation from even the lowest ($l = 0$) quasi-radial mode \cite{Stergioulas2003,Passamonti2006} making them potentially relevant f | or gravitational wave astronomy.
\section{Equations of state}
\label{sec2}
\subsection{Hadronic phase under a magnetic field}
\label{sec:A}
In this section we present an overview of the hadronic equations of state (EOS) used in this work. We describe |
hadronic matter within the framework of the relativistic non-linear Walecka (NLW) model \cite{WALECKA1986}. In this model we employ a field-theoretical approach in which the baryons interact via the exchange of $\sigma-\omega-\rho$ mesons in the presence | of a magnetic field $B$ along the $z-$axis. The total lagrangian density reads:
\begin{equation}\label{lt}
\mathcal{L}_{H}=\sum_{b}\mathcal{L}_{b}+\mathcal{L}_{m}+\sum_{l}\mathcal{L}_{l}+\mathcal{L}_B\,.
\end{equation}
where $\mathcal{L}_{b}$, $\mathca |
l{L}_{m}$, $\mathcal{L}_{l}$ and $\mathcal{L}_{B}$ are the baryons, mesons, leptons and electromagnetic field Lagrangians, respectively, and are given by
\begin{eqnarray}
\mathcal{L}_{b} &=& \overline{\psi}_{b}\left(i\gamma_{\mu}\partial^{\mu}-q_{b}\ga | mma_{\mu}A^{\mu}-m_{b}+g_{\sigma b}\sigma \right. \nonumber \\
&& \left. -g_{\omega b}\gamma_{\mu}\omega^{\mu}-g_{\rho b}\tau_{3b}\gamma_{\mu}\rho^{\mu}\right)\psi_{b}\,, \\
\mathcal{L}_{m} &=& \tfrac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma- |
m_{\sigma}^{2}\sigma^{2})-U(\sigma)+
\tfrac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu} \nonumber \\
&& -\tfrac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}+
\tfrac{1}{2}m_{\rho}^{2}\vec{\rho}_{\mu}\cdot\vec{\rho}_{\mu}-\tfrac{1}{4}P^{\mu\nu}P_{\mu\nu} \ | ,, \\
\mathcal{L}_{l} &=& \overline{\psi}_{l}\left(i\gamma_{\mu}\partial^{\mu}-q_{l}\gamma_{\mu}A^{\mu}-m_{l}\right)\psi_{l} \,, \\
\mathcal{L}_{B} &=& -\tfrac{1}{4}F^{\mu\nu}F_{\mu\nu} \,.
\end{eqnarray}
where he $b$-sum runs over the baryonic |
octet $b\equiv N~(p,~n),~\Lambda,~\Sigma^{\pm,0},~\Xi^{-,0}$, $\psi_{b}$ is the corresponding baryon Dirac field, whose interactions are mediated by the $\sigma$ scalar, $\omega_{\mu}$ isoscalar-vector and $\rho_{\mu}$ isovector-vector meson fields. The b | aryon charge, baryon mass and isospin projection are denoted by $q_{b}$, $m_{b}$ and $\tau_{3b}$, respectively, and the masses of the mesons are $ m_{\sigma}= 512~$ MeV, $m_{\omega}=783~$MeV and $m_{\rho}=770~$MeV. The strong interaction couplings of the n |
ucleons with the meson fields are denoted by $g_{\sigma N}=8.910$, $g_{\omega N}=10.610$ and $g_{\rho N}=8.196$. We consider that the couplings of the hyperons with the meson fields are fractions of those of the nucleons, defining $g_{iH}=X_{iH}g_{iN}$, w | here the values of $X_{iH}$ are chosen as $X_{\sigma H}=0.700$ and $X_{\omega H}=X_{\rho H}=0.783$ \cite{GLENDENNING2000}.
The term $U(\sigma)=\frac{1}{3}\,bm_{n}(g_{\sigma N}\sigma)^{3}-\frac{1}{4}\,c(g_{\sigma N}\sigma)^{4}$ denotes the scalar self-inte |
ractions \cite{NPA292:413:1977,PLB114:392:1982,AJ293:470:1985}, with
$c=-0.001070$ and $b=0.002947$. The mesonic and electromagnetic field tensors are given by their usual expressions $\Omega_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$ | , ${\bf P}_{\mu\nu}=\partial_{\mu}\vec{\rho}_{\nu}-\partial_{\nu}\vec{\rho}_{\mu}-g_{\rho b}(\vec{\rho}_{\mu}\times\vec{\rho}_{\nu})$ and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. The $l$-sum runs over the two lightest leptons $l\equiv e,\m |
u$ and $\psi_{l}$ is the lepton Dirac field.
The symmetric nuclear matter properties at saturation density adopted in this work are given by the GM1 parametrization \cite{PRL67:2414:1991}, with compressibility $K=300$ MeV, binding energy $B/A=-16.3$ MeV, s | ymmetry energy $a_{sym}=32.5$ MeV, slope $L=94$ MeV, saturation density $\rho_{0}= 0.153$ fm$^{-3}$ and nucleon mass $m=938$ MeV.
The following equations present the scalar and vector densities for the charged and uncharged baryons \cite{JPG36:115204:2009 |
,PRC89:015805:2014}, respectively:
\begin{eqnarray}
&\label{densities1}\rho_{b}^{s}=\frac{|q_{b}|B \bar{m}_{b} }{2\pi^{2}}\sum_{\nu}^{\nu_{\mathrm{max}}}\sum_{s}\frac{\bar{m}_{b}^{c}}{\sqrt{ \bar{m}_{b}^2 + 2\nu |q_{b}|B}} \ln\bigg|\frac{k_{F,\nu,s}^{ | \,b}+E_{F}^{\,b}}{\bar{m}_{b}^{c}} \bigg|,\\
&\label{densities2}\rho_{b}^{v}=\frac{|q_{b}|B}{2\pi^{2}}\sum_{\nu}^{\nu_{\mathrm{max}}}\sum_{s}k_{F,\nu,s}^{\,b},\\
&\label{densities3}\rho_{b}^{s}=\frac{\bar{m}_{b}}{4\pi^{2}}\sum_{s}\bigg[E_{F}^{\,b}k_{F,s}^{ |
\,b}-\bar{m}_{b}^{2}\ln\bigg|\frac{k_{F,s}^{\,b}+E_{F}^{\,b}}{\bar{m}_{b}}\bigg|\bigg] ,\\
&\label{densities4}\rho_{b}^{v}=\frac{1}{2\pi^{2}}\sum_{s}\bigg[\frac{1}{3}(k_{F,s}^{\,b})^{3}\bigg] ,
\end{eqnarray}
%
where $\bar{m}_{b}=m_{b}-g_{\sigma}\sigma$ | and $\bar{m}_{b}^{c}=\sqrt{ \bar{m}_{b}^2 + 2\nu |q_{b}|B}$. $\nu=n+\frac{1}{2}-$sgn$(q_{b})\frac{s}{2}=0,1,2,...$ are the Landau levels for the fermions with electric charge $q_{b}$, $s$ is the spin and assumes values
$+1$ for spin up and $-1$ for spin d |
own cases.
The energy spectra for the baryons are given by \cite{JPG35:125201:2008,APJ537:351:2000}:
\begin{eqnarray}\label{energy}
E_{\nu,s}^{\,b}=\sqrt{(k_{z}^{\,b})^{2}+\bar{m}_{b}^{2}+2\nu |q_{b}|B}+g_{\omega b}\omega^{0}+\tau_{3b}g_{\rho b}\rho^{0}\ | \
E_{s}^{\,b}=\sqrt{(k_{z}^{\,b})^{2}+\bar{m}_{b}^{2}+(k_{\perp}^{\,b})^{2}}+g_{\omega b}\omega^{0}+\tau_{3b}g_{\rho b}\rho^{0},
\end{eqnarray}
where $k_{\perp}^{\,b}=k_{x}^{\,b}+k_{y}^{\,b}$. The Fermi momenta $k_{F,\nu,s}^{\,b}$ of the charged baryons an |
d $k_{F,s}^{\,b}$ of the uncharged baryons and their relationship with the Fermi energies of the charged baryons $E_{F,\nu,s}^{\,b}$ and uncharged baryons $E_{F,s}^{\,b}$ can be written as:
\begin{eqnarray}\label{Momentum}
&&(k_{F,\nu,s}^{\,b})^{2}=(E_{F,\ | nu,s}^{\,b})^{2}-(\bar{m}_{b}^{c})^{2} \\
&&(k_{F,s}^{\,b})^{2}=(E_{F,s}^{\,b})^{2}-\bar{m}_{b}^{2}.
\end{eqnarray}
For the leptons, the vector density is given by:
\begin{eqnarray}\label{vector_{density_{leptons}}}
\rho_{l}^{v}=\frac{|q_{l}|B}{2\pi^{2}}\ |
sum_{\nu}^{\nu_{\mathrm{max}}}\sum_{s}k_{F,\nu,s}^{\,l},
\end{eqnarray}
where $k_{F,\nu,s}^{\,l}$ is the lepton Fermi momentum, which is related to the Fermi energy $E_{F,\nu,s}^{\,l}$ by:
\begin{eqnarray}\label{Momentuml}
(k_{F,\nu,s}^{\,l})^{2}=(E_{F,\n | u,s}^{\,l})^{2}-\bar{m}_{l}^{2}\,, \qquad l=e,\mu,
\end{eqnarray}
\noindent with $\bar{m}_{l}=m_{l}^{2}+2\nu |q_{l}|B$. The summation over the Landau level runs until $\nu_{\mathrm{max}}$; this is the largest value of $\nu$ for which the square of Fermi |
momenta of the particle is still positive and corresponds to the closest integer, from below to:
\begin{eqnarray}
&&\nu_{\mathrm{max}}=\bigg[\frac{(E_{F}^{\,l})^{2}-m_{l}^{2}}{2|q_{l}|B}\bigg], \quad\mathrm{leptons}\label{ll1}\\
&&\nu_{\mathrm{max}}=\bigg[ | \frac{(E_{F}^{\,b})^{2}-\bar{m}_{b}^{2}}{2|q_{b}|B}\bigg], \quad\mathrm{charged~baryons}.\label{ll2}
\end{eqnarray}
The chemical potentials of baryons and leptons are:
\begin{eqnarray}\label{chemicalp}
&&\mu_{b}=E_{F}^{\,b}+g_{\omega b}\omega^{0}+\tau_{3b |
}g_{\rho b}\rho^{0},\\
&&\mu_{l}=E_{F}^{\,l}=\sqrt{(k_{F,\nu,s}^{\,l})^{2}+m_{l}^{2}+2\nu |q_{l}|B}\,.
\end{eqnarray}
From the Lagrangian density~(\ref{lt}), and mean-field approximation, the energy density is given by
\begin{eqnarray}\label{energym}
\va | repsilon_{m}= & & \sum_{b}(\varepsilon_{b}^c + \varepsilon_{b}^n) +\tfrac{1}{2}m_{\sigma}\sigma_{0}^{2}\nonumber\\
&& +U(\sigma)+\tfrac{1}{2}m_{\omega}\omega_{0}^{2}+\tfrac{1}{2}m_{\rho}\rho_{0}^{2}\,,
\end{eqnarray}
where the expressions for the energy de |
nsities of charged baryons $\varepsilon_{b}^{c}$ and neutral baryons $\varepsilon_{b}^{n}$ are, respectively, given by:
\begin{eqnarray}\label{energy-densities-baryons}
\varepsilon_{b}^{c}& = &\frac{|q_{b}|B}{4\pi^{2}}\sum_{\nu}^{\nu_{\mathrm{max}}}\sum_{s | }\bigg[k_{F,\nu,s}^{\,b}E_{F}^{\,b} \nonumber \\
& & + (\bar{m}_{b}^{c})^{2}\ln\bigg|\frac{k_{F,\nu,s}^{\,b}+E_{F}^{\,b}}{\bar{m}_{b}^{c}}\bigg|\bigg],\label{ea1} \\
\varepsilon_{b}^{n}& = &\frac{1}{4\pi^{2}}\sum_{s}\bigg[\tfrac{1}{2} k_{F,\nu,s}^{\,b}(E |
_{F}^{\,b})^{3} - \tfrac{1}{4}\bar{m}_{b} \bigg(\bar{m}_{b}k_{F,\nu,s}^{\,b}E_{F}^{\,b} \nonumber \\
&& + \bar{m}_{b}^{3}\ln\bigg|\frac{E_{F}^{\,b}+k_{F,\nu,s}^{\,b}}{\bar{m}_{b}}\bigg|\bigg)\bigg]\, . \label{ea2}
\end{eqnarray}
\noindent The expression | for the energy density of leptons $\varepsilon_{l}$ reads
\begin{equation}
\varepsilon_{l}= \frac{|q_{l}|B}{4\pi^{2}}\sum_{l}\sum_{\nu}^{\nu_{\mathrm{max}}}\sum_{s}\bigg[k_{F,\nu,s}^{\,l}E_{F}^{\,l}+
\bar{m}_{l}^{2}\ln\bigg|\frac{k_{F,\nu,s}^{\,l}+E_{F |
}^{\,l}}{\bar{m}_{l}}\bigg|\bigg]\,. \label{ea3}
\end{equation}
The pressures of baryons and leptons are:
\begin{eqnarray}\label{pressurem}
P_{m}&=&\mu_{n}\sum_{b}\rho_{b}^{v}-\varepsilon_{m}, \\
P_{l}&=&\sum_{l}\mu_{l}\rho_{l}^{v}-\varepsilon_{l},
\end{ | eqnarray}
where the expression of the vector densities $\rho_{b}^{v}$ and $\rho_{l}^{v}$ are given in~(\ref{densities2}) and (\ref{vector_{density_{leptons}}}), respectively.
\subsection{Density-dependent magnetic field}
\label{B-density-depndence}
W |
e assume that the magnetic field $B$ in the EOS depends on the density according to \cite{PRL79:2176:1997,ChJAA3:359:2003,PRC80:065805:2009,JPG36:115204:2009,BJP42:428:2012}
\begin{equation}\label{cmdd}
B\left(\frac{\rho}{\rho_{0}}\right)=B_{\mathrm{ | surf}}+B_{0}\left\{ 1-\mathrm{exp}\left[ -\beta\left( \frac{\rho}{\rho_{0}}\right)^{\gamma} \right] \right\}\,,
\end{equation}
where $\rho=\sum_{b}\rho_{b}^{v}$ is the baryon density, $\rho_{0}$ is the saturation density, $B_{\mathrm{surf}}$ is the magnet |
ic field on the surface of a magnetar, taken equal to $10^{15}$~G in agreement with observational values, and $B_{0}$ is the magnetic field for larger densities. The parameters $\beta$ and $\gamma$ are chosen to reproduce two behaviors of the magnetic fiel | d: a fast decay with $\gamma=3.00$ and $\beta=0.02$ and a slow decay with $\gamma=2.00$ and $\beta=0.05$ \cite{PRC89:015805:2014}. According to the discussion in the previous subsection, we use two values for the magnetic field $B_{0}$, namely $10^{17}$~G |
and $3.1\times10^{18}$~G.
\subsection{On the isotropy of the pressure}
Notice that in the previous subsections we assumed that the matter pressure ($P_m + P_l$) is isotropic in spite of the high values of the magnetic field. As it has been shown i | n Ref. \cite{JPG41:015203:2014}, the anisotropic effects around $3.1\times10^{18}$~G are small, thus we restrict ourselves to magnetic fields below this value.
However, the purely field-related pressure $P_B \sim B^2$ may become dominant in the core of |
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